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if x and y are integers and y=|x+3| +|4-x|, does y equal 7

If we carefully look at the question we can see that "does y equal 7" actually means that y doesn't depend on x and then you can see that if we open two moduli with same signs (++ or --), x and -x disappear and 3+4 is 7. So, let's see when we can open moduli ++ or --:

++) x>-3 and x<4 or x e (-3,4) --) x<-3 and x>4 - it can't be.

So, if x between -3 and 4, y =7. Now, look at our statements: it is obvious that we need two statements.

\(y=|x+3|+|4-x|\) two check points: \(x=-3\) and \(x=4\) (check point: the value of \(x\) when expression in || equals to zero), hence three ranges to consider:

A. \(x<{-3}\) --> \(y=| x + 3| +|4-x| =-x-3+4-x=-2x+1\), which means that when \(x\) is in the range {-infinity,-3} the value of \(y\) is defined by \(x\) (we would have multiple choices of \(y\) depending on \(x\) from the given range);

B. \(-3\leq{x}\leq{4}\) --> \(y=|x+3|+|4-x|=x+3+4-x=7\), which means that when \(x\) is in the range {-3,4} the value of \(y\) is \(7\) (value of y does not depend on value of \(x\), when \(x\) is from the given range);

C. \(x>{4}\) --> \(y=|x+3|+|4-x|=x+3-4+x=2x-1\), which means that when \(x\) is in the range {4, +infinity} the value of \(y\) is defined by \(x\) (we would have multiple choices of \(y\) depending on \(x\) from the given range).

Hence we can definitely conclude that \(y=7\) if \(x\) is in the range {-3,4}

(1) \(x<4\) --> not sufficient (\(x<4\) but we don't know if it's \(\geq{-3}\)); (2) \(x>-3\) --> not sufficient (\(x>-3\) but we don't know if it's \(\leq{4}\));

(1)+(2) \(-3<x<4\) exactly the range we needed, so \(y=7\). Sufficient.

Answer: C.

OR: looking at \(y=|x+3|+|4-x|\) you can notice that \(y=7\) (\(y\) doesn't depend on the value of \(x\)) when \(x+3\) and \(4-x\) are both positive, in this case \(x-es\) cancel out each other and we would have \(y=|x+3|+|4-x|=x+3+4-x=7\). Both \(x+3\) and \(4-x\) are positive in the range \(-3<{x}<4\) (\(x+3>0\) --> \(x>-3\) and \(4-x>0\) --> \(x<4\)).

Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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26 Sep 2010, 22:09

4

This post received KUDOS

Let's first solve |x+3|+|4-x|=7 to answer "when is this true ?"

You can solve algebraically but it is much easier to do it using a simple number line approach. Remember |x-a| means distance between x and a on the number line Here the two points in question are -3 and 4 Now it is easy to imagine the three cases that x is to the left of -3, between -3 and 4 and to the right of 4. The only case when the two distances add up to the distance between -3 and 4, ie, 7 is case two. In case 1 and 3, the sum will exceed 7

1) could mean case 2 or 3. Not sufficient 2) could mean case 1 or 2. Not sufficient 1+2) can only mean case 2. Sufficient to know that y=7

Guys, the be low is my approcah for any modulus qtn in GMAT.

Remember. The meaning of |x-y| is "On the number line, the distance of X from +Y" The meaning of |x+y| is "On the number line, the distance of X from -Y" The meaning of |x| is "On the number line, the distance of X from 0".

Qtn:

for integers X and Y, If y=|x+3| + |4-x|, does y equals 7 ==> is the SUM of the distance b/e x and -3 , and x and 4 equals to 7? ==> .........-3......0...........4..... observe that X has to be anywhere b/w -3 and 4 or on any of these points for the total distance to be 7

Stmnt1: X < 4: Answer could be Yes if X is < 4 and b/w -3 and 4 but from the given information (i.e X<4) X could be some where left to -3 in which case the total distance would be > 7 hence insufficient.

......X....-3......0..........4 answer to the qtn: NO or ............-3...X...0.........4 answer to the qtn: YES

Stmnt2: X > -3: Answer could be Yes if X is > -3 and b/w -3 and 4 but from the given information (i.e X>-3) X could be some where right to 4 in which case the total distance would be > 7 hence insufficient. ..........-3......0..........4...X... answer to the qtn: NO or ..........-3...X...0.........4 answer to the qtn: YES

1&2

X must be b/w -3 and 4 ..........-3.X.X.X...0.X.X.X.X.X...4...... answer is always YES..hence Sufficient.

(1) If x = 3, y = |3+3| + |4-3| = 7. If x = -100, then y = 97 + 104 = 201. (2) If x = -2, y = 1 + 6 = 7. If x = 100, then y = 103 + 96 = 199.

Putting them together, you can quickly check every integer value of x from -3 to 4 and see that y = 7 for every one. It's only 6 values to check, you can do it very quickly in your head.

Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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26 May 2012, 07:38

At bunuel,

A. x<{-3} --> y=| x + 3| +|4-x| =-x-3+4-x=-2x+1, which means that when x is in the range {-infinity,-3} the value of y is defined by x (we would have multiple choices of y depending on x from the given range);

B. -3\leq{x}\leq{4} --> y=|x+3|+|4-x|=x+3+4-x=7, which means that when x is in the range {-3,4} the value of y is 7 (value of y does not depend on value of x, when x is from the given range);

C. x>{4} --> y=|x+3|+|4-x|=x+3-4+x=2x-1, which means that when x is in the range {4, +infinity} the value of y is defined by x (we would have multiple choices of y depending on x from the given range).

I understand the how you got the check points -3 and 4 but I am having a hard time understanding how to decide sign for x when you are removing absolute value symbol for example for (A) x<-3 y=| x + 3| +|4-x| how did you decide sign of "x" here ===> -x-3+4-x -2x+1

A. x<{-3} --> y=| x + 3| +|4-x| =-x-3+4-x=-2x+1, which means that when x is in the range {-infinity,-3} the value of y is defined by x (we would have multiple choices of y depending on x from the given range);

B. -3\leq{x}\leq{4} --> y=|x+3|+|4-x|=x+3+4-x=7, which means that when x is in the range {-3,4} the value of y is 7 (value of y does not depend on value of x, when x is from the given range);

C. x>{4} --> y=|x+3|+|4-x|=x+3-4+x=2x-1, which means that when x is in the range {4, +infinity} the value of y is defined by x (we would have multiple choices of y depending on x from the given range).

I understand the how you got the check points -3 and 4 but I am having a hard time understanding how to decide sign for x when you are removing absolute value symbol for example for (A) x<-3 y=| x + 3| +|4-x| how did you decide sign of "x" here ===> -x-3+4-x -2x+1

Similarly can u also explain for (B) and (C)

thank you!

-K

Absolute value properties: When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|\leq{-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);

When \(x\geq{0}\) then \(|x|=x\), or more generally when \(some \ expression\geq{0}\) then \(|some \ expression|\leq{some \ expression}\). For example: \(|5|=5\);

So, for example if \(x<-3\) then \(x+3<0\) and \(4-x>0\) which means that \(|x+3|=-(x+3)\) and \(|4-x|=4-x\) --> \(|x+3|+|4-x|=-(x+3)+4-x=-2x+1\).

\(y=|x+3|+|4-x|\) two check points: \(x=-3\) and \(x=4\) (check point: the value of \(x\) when expression in || equals to zero), hence three ranges to consider:

A. \(x<{-3}\) --> \(y=| x + 3| +|4-x| =-x-3+4-x=-2x+1\), which means that when \(x\) is in the range {-infinity,-3} the value of \(y\) is defined by \(x\) (we would have multiple choices of \(y\) depending on \(x\) from the given range);

B. \(-3\leq{x}\leq{4}\) --> \(y=|x+3|+|4-x|=x+3+4-x=7\), which means that when \(x\) is in the range {-3,4} the value of \(y\) is \(7\) (value of y does not depend on value of \(x\), when \(x\) is from the given range);

C. \(x>{4}\) --> \(y=|x+3|+|4-x|=x+3-4+x=2x-1\), which means that when \(x\) is in the range {4, +infinity} the value of \(y\) is defined by \(x\) (we would have multiple choices of \(y\) depending on \(x\) from the given range).

Hence we can definitely conclude that \(y=7\) if \(x\) is in the range {-3,4}

(1) \(x<4\) --> not sufficient (\(x<4\) but we don't know if it's \(\geq{-3}\)); (2) \(x>-3\) --> not sufficient (\(x>-3\) but we don't know if it's \(\leq{4}\));

(1)+(2) \(-3<x<4\) exactly the range we needed, so \(y=7\). Sufficient.

Answer: C.

OR: looking at \(y=|x+3|+|4-x|\) you can notice that \(y=7\) (\(y\) doesn't depend on the value of \(x\)) when \(x+3\) and \(4-x\) are both positive, in this case \(x-es\) cancel out each other and we would have \(y=|x+3|+|4-x|=x+3+4-x=7\). Both \(x+3\) and \(4-x\) are positive in the range \(-3<{x}<4\) (\(x+3>0\) --> \(x>-3\) and \(4-x>0\) --> \(x<4\)).

Hope it's clear.

inequalities are posing problems! - one doubt - when it is said "-3<x<4", in this range shouldn't we check -2, -1 or 2,1 etc and see what is the value for y? is y independent of x when x<0?

\(y=|x+3|+|4-x|\) two check points: \(x=-3\) and \(x=4\) (check point: the value of \(x\) when expression in || equals to zero), hence three ranges to consider:

A. \(x<{-3}\) --> \(y=| x + 3| +|4-x| =-x-3+4-x=-2x+1\), which means that when \(x\) is in the range {-infinity,-3} the value of \(y\) is defined by \(x\) (we would have multiple choices of \(y\) depending on \(x\) from the given range);

B. \(-3\leq{x}\leq{4}\) --> \(y=|x+3|+|4-x|=x+3+4-x=7\), which means that when \(x\) is in the range {-3,4} the value of \(y\) is \(7\) (value of y does not depend on value of \(x\), when \(x\) is from the given range);

C. \(x>{4}\) --> \(y=|x+3|+|4-x|=x+3-4+x=2x-1\), which means that when \(x\) is in the range {4, +infinity} the value of \(y\) is defined by \(x\) (we would have multiple choices of \(y\) depending on \(x\) from the given range).

Hence we can definitely conclude that \(y=7\) if \(x\) is in the range {-3,4}

(1) \(x<4\) --> not sufficient (\(x<4\) but we don't know if it's \(\geq{-3}\)); (2) \(x>-3\) --> not sufficient (\(x>-3\) but we don't know if it's \(\leq{4}\));

(1)+(2) \(-3<x<4\) exactly the range we needed, so \(y=7\). Sufficient.

Answer: C.

OR: looking at \(y=|x+3|+|4-x|\) you can notice that \(y=7\) (\(y\) doesn't depend on the value of \(x\)) when \(x+3\) and \(4-x\) are both positive, in this case \(x-es\) cancel out each other and we would have \(y=|x+3|+|4-x|=x+3+4-x=7\). Both \(x+3\) and \(4-x\) are positive in the range \(-3<{x}<4\) (\(x+3>0\) --> \(x>-3\) and \(4-x>0\) --> \(x<4\)).

Hope it's clear.

MOD questions always floor me. Could you please suggest some good material on MODs?

\(y=|x+3|+|4-x|\) two check points: \(x=-3\) and \(x=4\) (check point: the value of \(x\) when expression in || equals to zero), hence three ranges to consider:

A. \(x<{-3}\) --> \(y=| x + 3| +|4-x| =-x-3+4-x=-2x+1\), which means that when \(x\) is in the range {-infinity,-3} the value of \(y\) is defined by \(x\) (we would have multiple choices of \(y\) depending on \(x\) from the given range);

B. \(-3\leq{x}\leq{4}\) --> \(y=|x+3|+|4-x|=x+3+4-x=7\), which means that when \(x\) is in the range {-3,4} the value of \(y\) is \(7\) (value of y does not depend on value of \(x\), when \(x\) is from the given range);

C. \(x>{4}\) --> \(y=|x+3|+|4-x|=x+3-4+x=2x-1\), which means that when \(x\) is in the range {4, +infinity} the value of \(y\) is defined by \(x\) (we would have multiple choices of \(y\) depending on \(x\) from the given range).

Hence we can definitely conclude that \(y=7\) if \(x\) is in the range {-3,4}

(1) \(x<4\) --> not sufficient (\(x<4\) but we don't know if it's \(\geq{-3}\)); (2) \(x>-3\) --> not sufficient (\(x>-3\) but we don't know if it's \(\leq{4}\));

(1)+(2) \(-3<x<4\) exactly the range we needed, so \(y=7\). Sufficient.

Answer: C.

OR: looking at \(y=|x+3|+|4-x|\) you can notice that \(y=7\) (\(y\) doesn't depend on the value of \(x\)) when \(x+3\) and \(4-x\) are both positive, in this case \(x-es\) cancel out each other and we would have \(y=|x+3|+|4-x|=x+3+4-x=7\). Both \(x+3\) and \(4-x\) are positive in the range \(-3<{x}<4\) (\(x+3>0\) --> \(x>-3\) and \(4-x>0\) --> \(x<4\)).

Hope it's clear.

Thanks for the great explanation . I got this answer when I tackled it using the same approach .. but got an E when I tried to tackle it in the +/- (x+3)=+/-(4-x) . I found y = +7 , -7 , 2x-1 , -2x+1 , and on plugging values that satisfy the 2 statements together it turned out to a range of -5 to 5 . Could you please guide ? Thanks a million
_________________

\(y=|x+3|+|4-x|\) two check points: \(x=-3\) and \(x=4\) (check point: the value of \(x\) when expression in || equals to zero), hence three ranges to consider:

A. \(x<{-3}\) --> \(y=| x + 3| +|4-x| =-x-3+4-x=-2x+1\), which means that when \(x\) is in the range {-infinity,-3} the value of \(y\) is defined by \(x\) (we would have multiple choices of \(y\) depending on \(x\) from the given range);

B. \(-3\leq{x}\leq{4}\) --> \(y=|x+3|+|4-x|=x+3+4-x=7\), which means that when \(x\) is in the range {-3,4} the value of \(y\) is \(7\) (value of y does not depend on value of \(x\), when \(x\) is from the given range);

C. \(x>{4}\) --> \(y=|x+3|+|4-x|=x+3-4+x=2x-1\), which means that when \(x\) is in the range {4, +infinity} the value of \(y\) is defined by \(x\) (we would have multiple choices of \(y\) depending on \(x\) from the given range).

Hence we can definitely conclude that \(y=7\) if \(x\) is in the range {-3,4}

(1) \(x<4\) --> not sufficient (\(x<4\) but we don't know if it's \(\geq{-3}\)); (2) \(x>-3\) --> not sufficient (\(x>-3\) but we don't know if it's \(\leq{4}\));

(1)+(2) \(-3<x<4\) exactly the range we needed, so \(y=7\). Sufficient.

Answer: C.

OR: looking at \(y=|x+3|+|4-x|\) you can notice that \(y=7\) (\(y\) doesn't depend on the value of \(x\)) when \(x+3\) and \(4-x\) are both positive, in this case \(x-es\) cancel out each other and we would have \(y=|x+3|+|4-x|=x+3+4-x=7\). Both \(x+3\) and \(4-x\) are positive in the range \(-3<{x}<4\) (\(x+3>0\) --> \(x>-3\) and \(4-x>0\) --> \(x<4\)).

Hope it's clear.

Thanks for the great explanation . I got this answer when I tackled it using the same approach .. but got an E when I tried to tackle it in the +/- (x+3)=+/-(4-x) . I found y = +7 , -7 , 2x-1 , -2x+1 , and on plugging values that satisfy the 2 statements together it turned out to a range of -5 to 5 . Could you please guide ? Thanks a million

Could you please elaborate the red part?
_________________

Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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27 Mar 2013, 08:37

Bunuel wrote:

Could you please elaborate the red part?

I mean that combining both statements x is between -3 and 4 , so I plugged all the values in this range in both 2x-1 and -2x+1 giving the range of Ys -5 to 5
_________________

I mean that combining both statements x is between -3 and 4 , so I plugged all the values in this range in both 2x-1 and -2x+1 giving the range of Ys -5 to 5

y is equal to 1-2x only when x<-3. y is equal to 2x-1 only when x>4.

When -3<x<4, then y is equal to 7.
_________________

Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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07 May 2013, 20:08

mario1987 wrote:

Hi guys, I would like to deeply understand how to deal with absolute value questions like the one attached. Thank you very much

The Expression given in the Question is Y = lx+3l + l4-xl .. & Question asks Is Y = 7 ????

Statement 1 :: x<4 ..... when we plugin any value of x less than 4 till -3 we will get a result as Y = 7 but below -3 ... Y will not be equal to 7 . Therefore, Insufficient.

Similarly, Statement :: 2 .... is insufficient.......

with 1+2 ..... we get the value of x in between 4 & -3 ...... i.e., -3<x<4 ....... For all the value of x in between -3 & 4 ... the value of

Y is always. 7 .. Therefore, Sufficient....

Hence, C ...........
_________________

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